Turing proposed a mathematical model based on chemical reactions and diffusion of substances throughout the xy-plane, called reaction-diffusion, to study the emergence of patterns from a homogeneous medium. Hammel and Prusinkiewicz used L-systems and reaction-diffusion in a one-dimensional medium to model the formation of sea shell patterns, and in an expanding one-dimensional medium to model heterocyst spacing in the cyanobacterium Anabaena catenula. They considered the formation of these two patterns as a continuous deterministic process neglecting the noise which is inherent to such a process.
The approach taken in this work is to stochastically model reaction-diffusion in a spatial, and possibly expanding, linear structure using L-systems. The stochastic simulation method for chemical reaction kinetics which was developed by Gillespie is used to study this model. On the basis of theoretical considerations, the L-system modelling language L+C is extended to include a stochastic rewriting strategy based on Gillespie's algorithm.
Mikolaj Cieslak. Stochastic Simulation of Pattern Formation: An Application of L-systems. M.Sc. thesis, University of Calgary, June 2006.
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